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Questions and Answers
What is the domain of the function $f(x) = \log_2 x$?
What is the domain of the function $f(x) = \log_2 x$?
- Negative real numbers
- All real numbers
- Non-real numbers
- Positive real numbers (correct)
What is the range of the function $f(x) = \log_5 x$?
What is the range of the function $f(x) = \log_5 x$?
- Non-real numbers
- Positive real numbers
- All real numbers (correct)
- Negative real numbers
What are the x-intercepts of the functions $f(x) = \log_3 x$ and $g(x) = \log_{10} x$?
What are the x-intercepts of the functions $f(x) = \log_3 x$ and $g(x) = \log_{10} x$?
- (1, 0) and (1, 0) (correct)
- (0, 1) and (0, 1)
- (0, 1) and (1, 0)
- None of the above
What is the y-intercept of the function $f(x) = \log_7 x$?
What is the y-intercept of the function $f(x) = \log_7 x$?
What is the vertical asymptote for the function $f(x) = \log_4 x$?
What is the vertical asymptote for the function $f(x) = \log_4 x$?
What is the domain of the function $f(x) = rac{1}{2} imes ext{log}_2 x$?
What is the domain of the function $f(x) = rac{1}{2} imes ext{log}_2 x$?
Which of the following is true about the function $f(x) = ext{log}_5 x$?
Which of the following is true about the function $f(x) = ext{log}_5 x$?
What is the y-intercept of the function $f(x) = 1.5 imes ext{log}_3 x$?
What is the y-intercept of the function $f(x) = 1.5 imes ext{log}_3 x$?
What is the vertical asymptote for the function $f(x) = ext{log}_{10} x$?
What is the vertical asymptote for the function $f(x) = ext{log}_{10} x$?
Which of the following is true about the function $f(x) = ext{log}_2 x$?
Which of the following is true about the function $f(x) = ext{log}_2 x$?
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Study Notes
Logarithm Functions Overview
- The domain of ( f(x) = \log_2 x ) is all positive real numbers, expressed as ( x > 0 ).
- The range of ( f(x) = \log_5 x ) encompasses all real numbers, represented as ( (-\infty, \infty) ).
x-intercepts of Logarithmic Functions
- The x-intercept of ( f(x) = \log_3 x ) occurs at ( x = 1 ) since ( \log_3 1 = 0 ).
- The x-intercept of ( g(x) = \log_{10} x ) is also at ( x = 1 ) for the same reason, ( \log_{10} 1 = 0 ).
y-intercepts of Logarithmic Functions
- The function ( f(x) = \log_7 x ) does not have a y-intercept since it only exists for ( x > 0 ).
- For ( f(x) = 1.5 \cdot \log_3 x ), the y-intercept cannot be defined as logarithmic functions do not cross the y-axis.
Vertical Asymptotes of Logarithmic Functions
- The vertical asymptote for ( f(x) = \log_4 x ) is at ( x = 0 ).
- The vertical asymptote for ( f(x) = \log_{10} x ) is also at ( x = 0 ).
Properties of Logarithmic Functions
- For ( f(x) = \log_5 x ), it is true that as ( x ) increases, ( f(x) ) increases without bound, and it approaches negative infinity as ( x ) approaches zero from the positive side.
- For ( f(x) = \log_2 x ), it is generally true that the function is increasing and never crosses the x-axis except at ( x = 1 ).
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